Please make the linguistic distinction when discussing these terms. Power over family life is referred to by '-lineal' suffixes and power over politics in the public sphere is referred to by '-archy' suffixes.

The distinction between private and public life is important in large communities. It seems probable to me that the distinction would be weaker in small communities, where family obligations and family-based authority are likely to be the root of all obligations and claims to authority.

ETA: I didn't mean for that to sound like a confident declaration. I'm really open to learning something here. Are the public and private spheres so distinct in smaller communities? I wouldn't expect them to be, but I don't have a lot of anthropological knowledge. But if my expectation is correct, then the -lineal/-archal distinction wouldn't be so relevant.